Locally finite groups containing a $2$-element with Chernikov centralizer

TL;DR

This paper proves that in a locally finite group with a 2-element having a Chernikov centralizer, if the involution's centralizer is nilpotent, then the group contains a soluble subgroup of finite index.

Contribution

It establishes a new link between Chernikov centralizers of 2-elements and the existence of soluble subgroups of finite index in locally finite groups.

Findings

If a 2-element has Chernikov centralizer, then the group has a soluble subgroup of finite index under certain conditions.

The nilpotency of the involution's centralizer implies the existence of a soluble subgroup of finite index.

The result extends understanding of the structure of locally finite groups with specific centralizer properties.

Abstract

Suppose that a locally finite group $G$ has a $2$-element $g$ with Chernikov centralizer. It is proved that if the involution in $\langle g\rangle$ has nilpotent centralizer, then $G$ has a soluble subgroup of finite index.